Probabilistic power flow computation using Liouville–Gaussian copula and nested cubature rule

This paper aims to develop a nested cubature rule for probabilistic power flow (PPF) computation. In the case that marginal distributions of PPF inputs are unknown, a polynomial cumulative distribution function (CDF) model is derived to recover distribution functions of PPF inputs. With a percentile matching method, parameters of the polynomial CDF model are easily obtained by solving a system of linear equations. If PPF inputs include correlated random variables, a Liouville–Gaussian copula is proposed to map the PPF problem to a homogeneously correlated normal space, then, a suitable elliptical copula or Archimedean copula is employed to relate PPF inputs to an independent standard normal vector, whereby it can reproduce the asymmetric dependence structure of PPF inputs. In order to accurately calculate statistical moments of PPF outputs, a nested cubature rule is developed in the framework of Kronecker product and Hadamard matrix, it can capture interactive uncertainties among PPF inputs, and has a computational burden linear with the number of PPF inputs. Finally, case studies are performed to check the polynomial CDF model and Liouville–Gaussian copula for fitting correlated PPF inputs, a PPF computation is conducted on IEEE 118-bus system to demonstrate the efficiency and accuracy of the nested cubature rule.